Fourier Transform

[RyanA1084]

Hi all, I had this problem for homework and it stumped me. It's too late to get points for it, but I'd like to know for future reference.

Find the Fourier transform F(w)=integral from -infinity to infinity of f(t)e^(i*w*t)dt

f(t)=e^(-t^2/a^2)

i=sqrt(-1) w=omega=constant a=constant

This looks sort of like a gaussian integral:

integral of e^(-a*x^2)dx=sqrt(pi/a)

but I couldn't see how to do it...

The answer given by the book is sqrt(pi)ae^(-a^2*w^2/4)

Anyone know how to do this??

[troutinthemilk]

I think the thing you want to do here is complete the square in the arguements of the exponential. In this expression:
-t^2/a^2 + iwt = -(t-b)^2 +c.
where you can find b and c. When you've done that you can change variables and you will have and expression for the Gaussian that you are familiar with.